# example of permutation matrix

Consider the matrix

 $P=\begin{pmatrix}0&1&0&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&0&1&0\end{pmatrix}$

that corresponds to permuting the columns of the identity matrix under the permutation $(1243)$ (i.e. (http://planetmath.org/Ie), the first column of the identity matrix is the second column of $P$, the second column of the identity matrix is the fourth column of $P$, etc.). Then $P$ is a permutation matrix.

We will consider what happens when we multiply a $4\times 4$ matrix by $P$. For example, let $A$ be the matrix

 $A=\begin{pmatrix}4&2&6&8\\ 1&3&5&7\\ 1&0&1&0\\ -1&-2&-3&-4\end{pmatrix}.$

Then

 $PA=\begin{pmatrix}0&1&0&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&0&1&0\end{pmatrix}\begin{pmatrix}4&2&6&8\\ 1&3&5&7\\ 1&0&1&0\\ -1&-2&-3&-4\end{pmatrix}=\begin{pmatrix}1&3&5&7\\ -1&-2&-3&-4\\ 4&2&6&8\\ 1&0&1&0\end{pmatrix}.$

We notice that $PA$ has the same rows as $A$. Moreover, the rows of $A$ are the rows of $PA$ permuted under $(1243)$: The first row of $PA$ is the second row of $A$, the second row of $PA$ is the fourth row of $A$, etc.

Title example of permutation matrix ExampleOfPermutationMatrix 2013-03-22 15:03:14 2013-03-22 15:03:14 Wkbj79 (1863) Wkbj79 (1863) 8 Wkbj79 (1863) Example msc 15A36